Editing Cantor set (section)

==Historical remarks==
[[File:Cantor dust in two dimensions iteration 2.svg|thumb|an image of the 2nd iteration of Cantor dust in two dimensions]][[File:Cantor dust in two dimensions iteration 4.svg|alt=an image of the 4th iteration of Cantor dust in two dimensions|thumb|an image of the 4th iteration of Cantor dust in two dimensions]]
Cantor introduced what we call today the Cantor ternary set <math>\mathcal C</math> as an example "of a [[Perfect set|perfect point-set]], which is not everywhere-dense in any interval, however small."<ref name=":1">{{Cite web |last=Cantor |first=Georg |date=2021 |title="Foundations of a general theory of sets: A mathematical-philosophical investigation into the theory of the infinite", English translation by James R Meyer |url=https://www.jamesrmeyer.com/infinite/cantor-grundlagen.html#Fn_22_a |access-date=2022-05-16 |website=www.jamesrmeyer.com |at=Footnote 22 in Section 10}}</ref><ref name=":2">{{Cite journal |last=Fleron |first=Julian F. |date=1994 |title=A Note on the History of the Cantor Set and Cantor Function |url=https://www.jstor.org/stable/2690689 |journal=Mathematics Magazine |volume=67 |issue=2 |pages=136–140 |doi=10.2307/2690689 |jstor=2690689 |issn=0025-570X}}</ref>  Cantor described <math>\mathcal C</math> in terms of ternary expansions, as "the set of all real numbers given by the formula: <math>z=c_1/3 +c_2/3^2 + \cdots + c_\nu/3^\nu +\cdots </math>where the coefficients <math>c_\nu</math> arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements."<ref name=":1" />

A topological space <math>P</math> is perfect if all its points are limit points or, equivalently, if it coincides with its [[Derived set (mathematics)|derived set]] <math>P'</math>. Subsets of the real line, like <math>\mathcal C</math>, can be seen as topological spaces under the induced subspace topology.<ref name=":0" />

Cantor was led to the study of derived sets by his results on uniqueness of [[Fourier series|trigonometric series]].<ref name=":2" /> The latter did much to set him on the course for developing an [[axiomatic set theory|abstract, general theory of infinite sets]].

[[Benoit Mandelbrot]] wrote much on Cantor dusts and their relation to [[Fractals in nature|natural fractals]] and [[statistical physics]].<ref name=":3">{{Cite book |last=Mandelbrot |first=Benoit B. |url=https://www.worldcat.org/oclc/36720923 |title=The fractal geometry of nature |date=1983 |isbn=0-7167-1186-9 |edition=Updated and augmented |location=New York |oclc=36720923}}</ref> He further reflected on the puzzling or even upsetting nature of such structures to those in the mathematics and physics community. In [[The Fractal Geometry of Nature]], he described how "When I started on this topic in 1962, everyone was agreeing that Cantor dusts are at least as monstrous as the [[Koch snowflake|Koch]] and [[Peano curve]]s," and added that "every self-respecting physicist was automatically turned off by a mention of Cantor, ready to run a mile from anyone claiming <math>\mathcal C</math> to be interesting in science."<ref name=":3" />